I've been trying to find an answer, but to no avail. Is what we call a perfect interval somewhat arbitrary? It seems as if the modern definition is "perfect under inversion". I know the other thing people say is that it is consonant, but I can't find a rigorous definition of consonance.

Is there a solid definition of perfect intervals, lying around somewhere I just can't find?

8 Answers
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Based on your reaction to other very good answers posted here already, your question seems to boil down to: "Why do humans innately feel that certain intervals are consonant". And so much so that they are willing to call them "perfect". Before getting to that question, let's look at why Western culture might consider them "perfect". My answer to your question will be rather freeform because the truth of the matter is there is not really good answer to your question outside the music theory-based explanations given above.

The modern Western music system has been inherited from some of the groundwork set by Pythagoras. It has been heavily modified to the point now that the modern 12-tone equal temperament we use now has the spirit of the original ideas from Pythagoras even if it differs greatly in many other ways. To Pythagoras, and possibly many Greeks at the time, certain intervals sounded very pleasing to the ear. Mathematically, these intervals are superparticular ratios [(n + 1)/n) or multiples [(x*n)/n]. For example, 4/3 is a superparticular ratio and 3/1 is a multiple. In other words, when the two frequencies resonate together and the ratio of the frequencies comes out in either of these forms many people in Western culture would agree they are pleasing. The perfect ratios display this quality in the best sense: 2/1 is an octave, 3/2 is a perfect fifth, and 4/3 is a perfect fourth. There is the least amount of conflict in the frequencies between the notes allowing for more complete symmetrical intersection between the waveforms. This is probably why Pythagoras liked these intervals - the Pythagoreans loved this kind of mathematical perfection. He liked it so much he tried to develop a tuning system out of it (Pythagorean Tuning) which ended being impossible without introducing a tuning error (the Pythagorean Comma).

I am not too clear on how Pythagoras's discoveries exactly carried over through time but his ideas were often used and cited by other musicologists through time. One example is Ptolemy who created scales based of Pythagorean tuning that included other less consonant intervals (thirds). What I am getting at here is that our assumption of the "perfect" intervals derives from the fact that the system's originator (and possibly his culture) deemed them to be perfect. It's hard to say why the name persisted through time but needless to say, thousands of tunings systems were developed after Pythagoras, most of which tried to preserve the perfect fifth, fourth, and the octave while allowing wiggle room for other intervals to fit together in the scales (I'm oversimplifying but that's the idea).

But is it pleasing to humans in general? That depends. Many cultures developed other systems that don't necessarily have this obsession with the perfect intervals or used many others equally. Other cultures (Persian music) have divided the octave into 53-tones, 24-tones (some forms of Indian music), and other divisions. One response to this is that the majority of non-Western cultures tended to develop music systems that were melodically complex: complex scales over a single droning note, but not harmonically complex like Western music. So perhaps they never needed to develop the notions of "perfect" in the first place. There is also the fact that in the modern era we have become increasingly attracted to dissonant or unusual forms of harmony. There is widespread interest in rock/metal which emphasizes distorting the sound wave to emphasis dissonant overtones (even if the intervals actually played are quite consonant). Dubstep is not exactly harmonically pleasing either but it is popular. Modern Jazz uses some complex and dissonant forms of harmony. A lot of 20th century classical music is also very dissonant. The question comes down to if it's a matter of taste, the unexpected (things that surprise us make things interesting, a change from regularity), culture/social norms, or if it's innate. There's also a difference between enjoying dissonant music and actually finding it pleasing. I love dissonant music but I don't really find it more "pleasing" than consonant music - I like it because it is jarring.

Music psychology and cognitive neuroscience has not come to a firm conclusion on this question. There have been a lot of studies on this topic but none are quite conclusive. One simple explanation is that evolutionarily, the human brain learned to find patterns and structure to apply semantic meaning. This means that we seek things that have regularity and predictability and attempt to assign meaning to things to help them to fit within these frameworks. Dissonant music deliberately goes outside predictable frequency ratios that line up, producing uneven sounds. Perhaps the aversion to these sounds is a by-product of the general manner in which the brain functions in the world.

But this is a post hoc explanation. Cognitive neuroscience has been asking these questions for a long time and modern advances in computational neuroscience may soon provide an answer. A simple look at this question can be found in this Nature article.

To summarize: We probably call it "perfect" because of Pythagoras and musicologists that came after him. We probably think it's "perfect" for cultural and social reasons. If it is really "perfect" to us innately is to be determined.

There's some good stuff in this answer, but the super particularratio does not correspond well to “perfect” intervals, as the major third (5:4) and minor third (6:5) have the same kind of ratio.
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Bradd SzonyeAug 1 '14 at 6:46

You're completely correct. The point I was trying to make was that the Pythagoreans recognized superparticular ratios as being consonant but did not extend this principle beyond the fourth harmonic. This goes back to what I was saying about modern Western music "inheriting" the idea of the consonance of 2:1, 3:2, and 4:3, from Pythagoras as a fixed state that tuning systems were to achieve.
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syntonicCAug 1 '14 at 15:06

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What's more interesting to me though is that 12-tet does not use any of the just intervals beyond the perfect ones (+/- 1-2 cents). The major third is off by 14 cents (from the 5-limit major third) in 12-tet but nobody really seems to notice unless they have been exposed to Just Intonation for long enough. This really makes me think it's not very innate but learned/cultural. Just my speculation though.
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syntonicCAug 1 '14 at 15:07

@syntonicC: I would suggest that a 4:5:6 major chord is to a 4:5.06:6 chord what a photograph of scenery is to a tracking-camera shot of that same scenery. Adding a little motion to the scene makes it much easier for the brain to separate out the items within it. I don't think people's preference for slightly-sharp thirds is just cultural--I think that the brain needs the changing phase relationships between notes of a chord in order to hear them cleanly as distinct notes.
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supercatAug 1 '14 at 16:21

@supercat: might be a point for purely-sawtooth-synths-music, but with acoustic instruments there's always enough "movement" regardless of intonation. I don't think anybody really has a preference for slightly-sharp thirds as such, they'll either not notice them at all or notice them as dissonant. However 1) that dissonance can have leading character 2) if you tune some of the thirds to just 5/4 but keep the same 12 pitch classes and try to play arbitrary music, you'll invariably have some intervals gotten way more dissonant, which is really the reason why 12-edo has been so successful.
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leftaroundaboutAug 2 '14 at 15:28

There are four types of perfect interval: perfect unison, perfect fourth, perfect fifth, and perfect octave.

These can be thought of as belonging to two groups. In the first group, all intervals of a unison or an octave are called perfect because the note is not changed. An octave is twice (or half) the frequency of the first note.

The second group includes the perfect fifth or perfect fourth. Actually, traditionally the fourth was not considered consonant. However, since the fifth is perfect, and the inversion of the fifth is a fourth, then the fourth is exactly the same thing as a fifth and must also be perfect. These notes add a very slight amount of coloring but not really enough to constitute a harmony.

Rather than using dissonance or consonance (somewhat subjective terms), I prefer to think about it as adding harmonic content or not.

Take any root note, and add as many unisons, octaves, and fifths (or fourths, but please not both, because now these two will conflict with each other), and you have no real harmony. The unisons and octaves do not add harmonic content because they're the same note as the root. And the fifth doesn't add harmonic content because it is the strongest overtone in the harmonic series. In a nutshell, if you play the root note C, you are also to some extent playing a G because the G is audibly present in the harmonic series of the root note C. Whenever anyone plays a C, they're also playing a G, because physics. So whether you then use your instrument to play a second G or not, the G is present inside of the C anyways.

So perfect intervals are those which are so consonant that they don't add any harmony.

Note: edited for clarity due to a number of comments asking for clarification.

Forgive me, I'm not very well read in musical literature. What do you mean by "add any harmony"?
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AnthonyJul 31 '14 at 7:08

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A harmony is when you combine two or more notes and they create a sound that none of the notes could have had by itself. If your first note is "C", adding the octave "C" or the perfect fifth "G" doesn't really create any harmony. But adding an "E" and an "A" to the "C" would add quite a bit of harmony. That's because those notes are not "C", and not "G" which as I mentioned is already contained inside of the C.
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GreyJul 31 '14 at 7:11

I think I might understand. But you say "whenever anyone plays a C, they're also playing a G, because physics." I'm not sure I understand what physics you're talking about, I feel as if whatever logic we use to "show" there is a G could also be used to "show" there is any other note.
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AnthonyJul 31 '14 at 7:16

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I think you're convoluting interval names and dissonance. There is nothing wrong with the term "perfect fourth". This is simply a fourth that is neither augmented nor diminished. Whether that is considered dissonance or consonance is simply another matter.
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Roland BoumanJul 31 '14 at 7:31

@Anthony It can. But most other notes would fall outside of your own ability to hear. The G is audible.
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GreyJul 31 '14 at 7:33

The rules seem to have been man-made. A second (the D) is the same note in major and minor, just like the 4th and 5th. All three are present in both major and minor keys, so it seems (to me), illogical to say that a 2nd can be major or minor, especially when a minor 2nd doesn't appear in a minor key ! Yes, it's all technicality, but seems artificial.What do you think ?
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TimJul 31 '14 at 8:07

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The rules are very much man-made. The consonances and resonances appear to exist in nature apart from human participation, but music is largely a construct of the mind interpreting the sounds it hears, and music theory tries to describe this after-the-fact. So the artificiality is rather par for the course. Not helping things is the fact that the terms major and minor are used to designate different things: the Major/Minor scales, major/minor intervals. The Major scale is composed of all major intervals, but the Minor scale is not all minor, that is the Phrygian mode.
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luser droogJul 31 '14 at 8:21

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It always makes me smile that a minor 6th chord has a major 6th in it...
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TimJul 31 '14 at 8:34

@Tim, by minor sixth chord, do you mean the first inversion of a minor triad? I don't have any issue with that. The precedence is the kind of triad (major, minor, diminished) and then the inversion - sixth being first inversion. That said there seem to be a lot of different chord naming schemes, and even more system to denote them.
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Roland BoumanJul 31 '14 at 9:02

@RolandBouman - a minor 6th chord is I-mIII-V-VI, as in C-Eb_G-A. A minor triad with an added major 6th.I'm not using classical inversion notation.
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TimJul 31 '14 at 9:24

All intervals can be turned upside down.(Called inverted). Thus a C-E as a major third, when played E-C becomes a minor sixth. There is a 'rule of nine'.Minors become majors, majors become minors, augmenteds become diminisheds, etc. The exceptions are the octaves, 4ths and 5ths. (Unison doesn't count !) Those do not change their identities. A 4th of C-F becomes a 5th of F-C, BUT, the interval stays as is - perfect. It hasn't changed.

Ugh, I keep finding this a little unsatisfactory. First, it depends on our definition of major and minor- which I suppose is fine, (although I'm not sure how to make that definition un-arbitrary.) Second, it doesn't seem enlightening in any way to me, as to why we called it a perfect- why is this invariance under inversion such a good quality?
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AnthonyJul 31 '14 at 17:16

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Don't forget the Tritone, which is the same even when inverted.
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KajiJul 31 '14 at 17:39

^Well sure, but thing is like, staying in the major scale under inversion, right? (I still have no idea why that is perfect.)
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AnthonyJul 31 '14 at 17:41

Basically, it's the fact that it doesn't change when it is in major. It doesn't even have to be in the major scale. It still is the same in minor. Perfect maybe is not a quality imbued upon the interval, just a name. When all this was labelled, the tritone was disallowed, as it was perceived as the Devil's interval. And the definition of major and minor are pre-determined, they are not open to jurisdiction.
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TimJul 31 '14 at 18:28

@Kaji Not exactly. C-F# is an augmented fourth. F#-C is a diminished fifth. Same interval, different name.
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John KugelmanJul 31 '14 at 23:00

A “perfect” interval is one that has nice small integer frequency ratios in Pythagorean tuning. These are traditionally considered the most consonant intervals.

P1 = 1:1

P8 = 2:1

P5 = 3:2

P4 = 4:3

Major and minor intervals have more complex ratios:

M2 = 9:8

m7 = 16:9

M6 = 27:16

m3 = 32:27

M3 = 81:64

m6 = 128:81

M7 = 243:128

m2 = 256:243

(They are distinguished by major intervals having a power of 3 in the numerator, and minor intervals having a power of 3 in the denominator.)

Augmented and diminished ratios, being father away from unison on the circle of fifths, are more complex still.

This classification may not make as much sense in other tuning systems like 5-limit just intonation, which aims to make major and minor thirds more consonant by simplifying their ratios to 5:4 and 6:5, or to the now-ubiquitous equal temperament which abandons integer ratios altogether. But musical terminology is slow to change.

All the rest have answered in terms of high-level music theory concepts, but I think it can be interesting to look at the intervals as raw coefficients instead. Harmonic intervals between notes are the intervals that can be expressed with simple rational numbers, where a "simple" rational number is one with a small amount of small prime factors.

For example, the distance between two tones (let's say, 440Hz and 880 Hz) is an octave if the frequency of the second tone is exactly two times the frequency of the first: 2 and 1/2 are the simplest rational numbers possible after the unison.

As our ear detects two tones that only differ by an octave as the "same" tone, multiplying or dividing by 2 an arbitrary number of times doesn't make intervals less simple. This makes 3 the simplest "significant" prime number. A fifth is an interval of 3/2, and a fourth is an interval of 2/3*, so we may conclude that a perfect interval is an interval that contains at most a single 3 as a prime factor and no other prime factor(as I said, we don't care about 2s).

* Technically, in the equally tempered scale this is not literally true: a fifth is 2^(7/12), which sliightly differs from 3/2, but our brain can't tell the difference.

@Anthony: See the Harmony section at en.wikipedia.org/wiki/Musical_acoustics for the math, but basically two frequencies such as 200Hz and its perfect fifth, 300Hz, have lots of complementary partials (partials for 300 = 300, 600, 900, 1200, etc.) so they resonate well with each other. Frequencies like 200Hz and 522Hz don't resonate as well.
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Kyle HaleJul 31 '14 at 20:06

It's only true for 12-tone equal temperament, which is close to having the pure 3/2 from Pythagorean tuning. (53-TET is closer, but poorly suited to piano-like instruments.) 19-TET has a better approximation to the 6/5 just minor third (2^(5/19) ≈ 1.2001), and 31-TET has a better approximation to the 5/4 just major third (2^(10/31) ≈ 1.2506).
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dan04Aug 1 '14 at 3:57